Supplementary Material for Integration of national demographic-disturbance relationships into local caribou population viability projections can reduce uncertainty and inform monitoring decisions

A Definitions of variables and parameters in the Bayesian population model

Table A1. Variables and parameters in the Bayesian population model and the national demographic model.

Symbol	Units	Description
Indices & constants
$t$	year	Year.
$m$	month	Month.
$i$	ID	Individual animal identifier.
Observed
$A_t$	%	Buffered anthropogenic disturbance.
$F_t$	%	Fire disturbance.
$\hat{J}_t$	animals	Calves observed in calf:cow survey.
$\hat{W}_t$	animals	Cows observed in calf:cow survey.
$\hat{T}_t$	animals	Number of collared cows.
$\hat{S}_t$	probability	Kaplan-Meier estimate of observed adult female survival.
$\tau_t$		Precision of Kaplan-Meier survival estimate.
$\hat{I}_{i,t,m}$	0 or 1	Indicator of whether animal $i$ is dead or alive.
Estimated state
$R_t$	rate	Recruitment (calves per cow).
$S_t$	probability	Adult female survival.
$N_t$	animals	Adult female population size.
$\bar{\lambda}_{y,t}$	rate	Average population growth over $y$ years.
$X_t$	rate	Recruitment adjusted for sex ratio and (optionally) delayed age of first reproduction.
$W_t$	animals	Surviving adult females before recruitment.
$J_t$	animals	New recruits to adult age class.
$\lambda_t$	rate	Annual population growth.
Estimated Parameters
$\beta^R_0$		Recruitment intercept.
$\beta^R_a$		Anthropogenic disturbance slope for recruitment.
$\beta^R_f$		Fire slope for recruitment.
$\sigma^2_{R}$		Variance in random effect of year on recruitment.
$\beta^S_0$		Survival intercept.
$\beta^S_a$		Anthropogenic disturbance slope for survival.
$\sigma^2_{S}$		Variance in random effect of year on survival.
True state
$\dot{R}_t$	rate	Recruitment (calves per cow).
$\dot{S}_t$	probability	Adult female survival.
$\dot{N}_t$	animals	Adult female population size.
$\dot{X}_t$	rate	Recruitment adjusted for sex ratio and (optionally) delayed age of first reproduction.
$\dot{W}_t$	animals	Surviving adult females before recruitment.
$\dot{J}_t$	animals	New recruits to adult age class.
True demographic parameters
$N_0=5000$	animals	Initial adult population size.
$\bar{R}_t$	rate	Expected recruitment (calves per cow).
$\bar{S}_t$	probability	Expected adult female survival.
$\dot{\beta}^R_0$		Recruitment intercept.
$\dot{\beta}^R_a$		Anthropogenic disturbance slope for recruitment.
$\dot{\beta}^R_f$		Fire slope for recruitment.
$\dot{\beta}^S_0$		Survival intercept.
$\dot{\beta}^S_a$		Anthropogenic disturbance slope for survival.
$p_0=1$		Maximum recruitment multiplier.
$p_k=0.6$		Recruitment multiplier at carrying capacity.
$k=100$		Carrying capacity multiplier.
$b=4$		Density dependence shape parameter.
$a=1$		Allee effect parameter.
$r_{max}=1.3$		Maximum population growth rate.
$\nu_R=0.46$		Coefficient of variation in $\dot{R}_t$ among years.
$\nu_S=0.08696$		Coefficient of variation in $\dot{S}_t$ among years.
$l_R=0$		Minimum value for $\dot{R}_t$.
$l_S=0$		Minimum value for $\dot{S}_t$.
$h_R=0$		Maximum value for $\dot{R}_t$.
$h_S=0$		Maximum value for $\dot{S}_t$.
Observation model parameters
$d=1,2,4,8,16, \text{ or } 24$	years	Monitoring duration.
$n=15,30, \text{ or } 60$	animals	Target number of collared females.
$o=1 \text{ or } 4$	years	Years between collar deployments.
$y=3$	years	Assessment period for population growth rate.
$w=3,6,\text{ or } 9$	ratio	The apparent number of adult females per collared animal in composition survey.
$\dot{q}=\text{uniform}(0,0.6)$	ratio	True ratio of young bulls to adult females in composition survey.
$\dot{u}=\text{uniform}(0,0.2)$	probability	True probability of misidentifying young bulls as adult females and vice versa in composition survey.
$\dot{z}=\text{uniform}(0,0.2)$	probability	True probability of missing calves in composition survey.
$\dot{c}$		True bias correction term in simulation model (Supplement B).
$q=\text{uniform}(0,0.6)$	ratio	Unknown ratio of young bulls to adult females in composition survey.
$u=\text{uniform}(0,0.2)$	probability	Unknown probability of misidentifying young bulls as adult females and vice versa in composition survey.
$z=\text{uniform}(0,0.2)$	probability	Unknown probability of missing calves in composition survey.
$c$		Unknown bias correction term in Bayesian model (Supplement B).
Expected value of perfect and sample information
$\psi=\text{viable or not}$	binary	True status of a population.
$\alpha=\text{viable or not}$	binary	Assigned population status.
$\nu_{\alpha,\psi}=1 \text{ when } \alpha=\psi, 0 \text{ otherwise}$	binary	Value of classification decision.
$x$		Simulated observations from an example population.
$\tau$		A disturbance and monitoring scenario.
$P(\psi\|\tau)$	probability	Prior probability of state $\psi$.
$P(\psi\|x,\tau)$	probability	Posterior probability of state $\psi$.
$EVPI_\tau$		Expected value of perfect information.
$EVSI_\tau$		Expected value of sample information.

B Model of bias in recruitment estimates from calf:cow surveys

Assume each group of animals in a calf:cow composition survey contains one or more collared adult females ($T$), and may also include: uncollared adult females misidentified as young bulls or unknown sex ($U$); correctly identified uncollared adult females ($V$); young bulls correctly identified as male or unknown sex ($O$); young bulls misidentified as uncollared adult females ($P$); observed calves ($J$); and unobserved calves ($K$). The apparent number of adult females in the group is $T+V+P=Tw$, where $w$ is a multiplier that defines the apparent number of adult females as a function of the number of collared animals. Our objective is to model the sex and bias-corrected recruitment rate $X=\frac{J+K}{2(T+U+V)}$ as a function of the observed calf:cow ratio $R=J/(T+V+P)$, the cow multiplier $w$, the ratio of young bulls to adult females $q$, and the misidentification probabilities $u$ and $z$. We start by solving for $T+U+V$ as a function of $q,w,u$ and $T$. Recognize that $P=Tw-T-V$, $U+V=V/(1-u)$, and $P+O=P/u$ to write $q$ as \[q=\frac{Tw-T-V}{uV/(1-u)}.\] Rearrange to get \[V=\frac{T(w-1)(1-u)}{qu+1-u}.\] Recognize that $U=Vu/(1-u)$ to write $T+U+V$ as a function of $q,w,u$ and $T$: \[T+U+V=T\frac{qu+w-u}{qu+1-u}.\] Recognize that the number of observed calves $J$ is the product of the apparent recruitment rate and the apparent number of adult females $J=RTw$, and that therefore $J+K=RTw/(1-z)$ to rewrite the bias corrected recruitment rate $X=\frac{J+K}{2(T+U+V)}$ as a function of $w,u,z$ and $R$: \[X=R\frac{w(1+qu-u)}{2(w+qu-u)(1-z)}.\] For simplicity, we write $X$ as a function of a bias correction term $c$: \[c=\frac{w(1+qu-u)}{(w+qu-u)(1-z)}; X=cR/2\]

If we also adjust for delayed age at first reproduction (DeCesare et al. 2012; Eacker et al. 2019), the adjusted recruitment rate becomes \[X=\frac{cR/2}{1+cR/2}.\]

Variation in the sex and bias corrected recruitment rate $X$ with the cow multiplier $w$, the ratio of young bulls to adult females $q$, and the adult misidentification probability $u$, and the probability of missing a calf $z$. If only collared females are counted ($w=1$) and there is a risk of missing calves $z>0$ then the apparent recruitment rate $R$ will be lower than $X$ (first column). When uncollared animals are counted ($w>1$), the risk of misidentifying adults $u$ inflates the apparent recruitment rate $R$ as on average more cows are misidentified as bulls than vice versa, but this effect is smaller than the effect of missing calves $z$. Increasing the proportion of bulls in the group $q$ reduces the effect of adult misidentification $u$.

Figure B.1: Variation in the sex and bias corrected recruitment rate $X$ with the cow multiplier $w$, the ratio of young bulls to adult females $q$, and the adult misidentification probability $u$, and the probability of missing a calf $z$. If only collared females are counted ($w=1$) and there is a risk of missing calves $z>0$ then the apparent recruitment rate $R$ will be lower than $X$ (first column). When uncollared animals are counted ($w>1$), the risk of misidentifying adults $u$ inflates the apparent recruitment rate $R$ as on average more cows are misidentified as bulls than vice versa, but this effect is smaller than the effect of missing calves $z$. Increasing the proportion of bulls in the group $q$ reduces the effect of adult misidentification $u$.

In the Bayesian model, uncertainty about the value of the bias correction term $c$ (grey bars) is approximated with a Log-normal distribution (red). In these examples, the ratio of young bulls to adult females $q$ varies uniformly between 0 and 0.6, the adult misidentification probability $u$ varies uniformly between 0 and 0.2, and the probability of missing a calf $z$ varies uniformly between 0 and 0.2. We assume that the apparent number of adult females per collared animal $w$ is known.

Figure B.2: In the Bayesian model, uncertainty about the value of the bias correction term $c$ (grey bars) is approximated with a Log-normal distribution (red). In these examples, the ratio of young bulls to adult females $q$ varies uniformly between 0 and 0.6, the adult misidentification probability $u$ varies uniformly between 0 and 0.2, and the probability of missing a calf $z$ varies uniformly between 0 and 0.2. We assume that the apparent number of adult females per collared animal $w$ is known.

C Calibration of Priors

We begin by setting random effects to zero, varying the prior standard deviations of the survival intercept (sIntSE is $\beta^S_0$ in Table 2) and recruitment intercept (rIntSE is $\beta^R_0$ in Table 2), and comparing the prior predictive intervals of the Bayesian model to the 95% predictive intervals from Johnson et al.’s (2020) regression analysis in a case with no anthropogenic disturbance (top row of Fig C.1). We set the prior standard deviation intercept parameters (sIntSE = 0.04, rIntSE = 0.25) so that the upper bounds of the Bayesian predictive intervals match the upper bounds of Johnson et al.’s predictive intervals.

We then vary the prior standard deviations of the slopes of the relationships between anthropogenic disturbance and demographic rates (sAnthroSlopeSE is $\beta^S_R$ and rAnthroSlopeSE is $\beta^R_R$ in Table 2), and compare the prior predictive intervals of the Bayesian model to the 95% predictive intervals from Johnson et al.’s (2020) regression analysis in a case with 90% anthropogenic disturbance (bottom row of Fig C.1). We set the prior standard deviation of the slopes (sAnthroSlopeSE = 0.0005, rAnthroSlopeSE = 0.006) so that the upper bound of the Bayesian predictive interval does not substantially exceed the upper bound of Johnson et al.’s predictive interval.

The choice of priors for the random effects of year on survival and recruitment are less important because the random effects have no effect on expected population growth rate (Figs C.2 and C.3). We set the hyperprior for the standard deviation of the survival random effect to vary uniformly between sSigmaMin = 0.00001 and sSigmaMax = 0.06 to exceed the variation among national model simulations by a moderate amount (Fig C.2). We set the hyperprior for the standard deviation of the recruitment random effect to vary uniformly between rSigmaMin = 0.00001 and rSigmaMax = 0.2 to exceed the variation among national model simulations by a moderate amount (Fig C.3).

Given intensive monitoring (15 years of 60 collars per year) of populations where demographic rates are higher or lower than the national average (from 97.5% and 2.5% quantiles of the national model), and the slope of the anthropogenic disturbance relationship differs from the national model, these priors are vague enough to allow posteriors to reflect local demographic rates (Figs C.4 and C.5).

Effect of variation in survival and recruitment prior standard deviations on the prior 95% predictive intervals and means of the Bayesian model (orange bands and lines) with no random effects (no interannual variation). Means and 95% predictive intervals from Johnson et al.'s 2020 regression model (black lines) summarize observed variation across the Canada.

Figure C.1: Effect of variation in survival and recruitment prior standard deviations on the prior 95% predictive intervals and means of the Bayesian model (orange bands and lines) with no random effects (no interannual variation). Means and 95% predictive intervals from Johnson et al.’s 2020 regression model (black lines) summarize observed variation across the Canada.

Differences between prior predictive survival distributions from the Bayesian model ('Bayesian') and simulated outcomes from the national model ('national'), assuming no local monitoring. In these examples anthropogenic disturbance is 0. Bands show 95% posterior predictive intervals ('local'), and the 2.5th and 97.5th percentiles for 1000 sample populations from the national model.

Figure C.2: Differences between prior predictive survival distributions from the Bayesian model (‘Bayesian’) and simulated outcomes from the national model (‘national’), assuming no local monitoring. In these examples anthropogenic disturbance is 0. Bands show 95% posterior predictive intervals (‘local’), and the 2.5th and 97.5th percentiles for 1000 sample populations from the national model.

Differences between prior predictive recruitment distributions from the Bayesian model ('Bayesian') and simulated outcomes from the national model ('national'), assuming no local monitoring. In these examples anthropogenic disturbance is 0. Bands show 95% posterior predictive intervals ('local'), and the 2.5th and 97.5th percentiles for 1000 sample populations from the national model.

Figure C.3: Differences between prior predictive recruitment distributions from the Bayesian model (‘Bayesian’) and simulated outcomes from the national model (‘national’), assuming no local monitoring. In these examples anthropogenic disturbance is 0. Bands show 95% posterior predictive intervals (‘local’), and the 2.5th and 97.5th percentiles for 1000 sample populations from the national model.

Posterior predicted survival from the Bayesian model fitted to 15 years of data, with 60 collars per year. Survival in these example populations is lower (sQuantile=0.025) or higher (sQuantile=0.975) than the national average, with no anthropogenic disturbance slope (sSlopeMod = 0), or an anthropogenic disturbance slope that is twice as steep as expected (sSlopeMod = 2). In these examples anthropogenic disturbance is 90%. Bands show 95% posterior predictive intervals, dots show true survival, and triangles show simulated observations.

Figure C.4: Posterior predicted survival from the Bayesian model fitted to 15 years of data, with 60 collars per year. Survival in these example populations is lower (sQuantile=0.025) or higher (sQuantile=0.975) than the national average, with no anthropogenic disturbance slope (sSlopeMod = 0), or an anthropogenic disturbance slope that is twice as steep as expected (sSlopeMod = 2). In these examples anthropogenic disturbance is 90%. Bands show 95% posterior predictive intervals, dots show true survival, and triangles show simulated observations.

Posterior predicted recruitment from the Bayesian model fitted to 15 years of data, with 60 collars per year. Recruitment in these example populations is lower (rQuantile=0.025) or higher (rQuantile=0.975) than the national average, with no anthropogenic disturbance slope (rSlopeMod = 0), or an anthropogenic disturbance slope that is twice as steep as expected (rSlopeMod = 2). In these examples anthropogenic disturbance is 90%. Bands show 95% posterior predictive intervals, dots show recruitment, and triangles show simulated observations.

Figure C.5: Posterior predicted recruitment from the Bayesian model fitted to 15 years of data, with 60 collars per year. Recruitment in these example populations is lower (rQuantile=0.025) or higher (rQuantile=0.975) than the national average, with no anthropogenic disturbance slope (rSlopeMod = 0), or an anthropogenic disturbance slope that is twice as steep as expected (rSlopeMod = 2). In these examples anthropogenic disturbance is 90%. Bands show 95% posterior predictive intervals, dots show recruitment, and triangles show simulated observations.

D Alternative disturbance and monitoring scenarios

Figure D.1: Static anthropogenic disturbance and monitoring scenarios.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroStatic) for static anthropogenic disturbance scenarios). In these examples there are 3 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.2: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.1 for static anthropogenic disturbance scenarios). In these examples there are 3 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroStatic) for static anthropogenic disturbance scenarios). In these examples there are 6 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.3: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.1 for static anthropogenic disturbance scenarios). In these examples there are 6 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroStatic) for static anthropogenic disturbance scenarios). In these examples there are 9 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.4: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.1 for static anthropogenic disturbance scenarios). In these examples there are 9 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

Figure D.5: Declining anthropogenic disturbance and monitoring scenarios.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroDeclining) for declining anthropogenic disturbance scenarios). In these examples there are 3 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.6: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.5 for declining anthropogenic disturbance scenarios). In these examples there are 3 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroDeclining) for declining anthropogenic disturbance scenarios). In these examples there are 6 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.7: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.5 for declining anthropogenic disturbance scenarios). In these examples there are 6 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

$The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure \@ref(fig:figDisturbanceAnthroDeclining) for declining anthropogenic disturbance scenarios). In these examples there are 9 cows per collared cow in recruitment surveys. See Figure 4 for additional details.$

Figure D.8: The probability of a correct status assessment varies with the amount of anthropogenic disturbance (columns), the number of years projected (rows), and monitoring effort (see Figure D.5 for declining anthropogenic disturbance scenarios). In these examples there are 9 cows per collared cow in recruitment surveys. See Figure 4 for additional details.

E Expected value of perfect information

Figure E.1: The expected value of perfect information (EVPI) varies among anthropogenic disturbance scenarios (see increasing, decreasing and stable scenarios in Figs 1, D.1 and D.5 respectively) and the number of years projected. When disturbance is high EVPI is low, indicating that local sampling will not alter status assessments. Increasing the forecast horizon can either increase or decrease the EVPI as the anthropogenic disturbance trend makes outcomes more or less predictable.

F Sensitivity to unkown bias in recruitment estimates from calf:cow surveys

Adding unknown unknown bias does increase uncertainty, but these increases are not large (compare Figs F.1/F.2 and F.3/F.4). A key reason for this result is that we assumed the same range of values for the probability of misclassifying adults, and the probability of missing calves (supplement B). Presuming that surveyed groups contain more cows than bulls, and that there is an equal probability of misclassifying cows and young bulls and vice versa, then misclassification errors cause high estimated recruitment. Missing calves causes low estimated recruitment, and together these errors tend to cancel one another out.

The probability of a correct status assessment with unknown bias in recruitment estimates from calf:cow surveys. In these examples there are 3 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

Figure F.1: The probability of a correct status assessment with unknown bias in recruitment estimates from calf:cow surveys. In these examples there are 3 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

The probability of a correct status assessment without bias in recruitment estimates from calf:cow surveys. In these examples there are 3 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

Figure F.2: The probability of a correct status assessment without bias in recruitment estimates from calf:cow surveys. In these examples there are 3 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

The probability of a correct status assessment with unknown bias in recruitment estimates from calf:cow surveys. In these examples there are 9 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

Figure F.3: The probability of a correct status assessment with unknown bias in recruitment estimates from calf:cow surveys. In these examples there are 9 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

The probability of a correct status assessment without bias in recruitment estimates from calf:cow surveys. In these examples there are 9 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

Figure F.4: The probability of a correct status assessment without bias in recruitment estimates from calf:cow surveys. In these examples there are 9 cows per collared cow in recruitment surveys, and disturbance increases over time (Figure 1). See Figure 4 for additional details.

G Colophon

This report was generated on 2024-06-17 12:17:57.88954 using the following computational environment and dependencies:

The current Git commit details are:

Symbol	Units	Description
Indices & constants
\(t\)	year	Year.
\(m\)	month	Month.
\(i\)	ID	Individual animal identifier.
Observed
\(A_t\)	%	Buffered anthropogenic disturbance.
\(F_t\)	%	Fire disturbance.
\(\hat{J}_t\)	animals	Calves observed in calf:cow survey.
\(\hat{W}_t\)	animals	Cows observed in calf:cow survey.
\(\hat{T}_t\)	animals	Number of collared cows.
\(\hat{S}_t\)	probability	Kaplan-Meier estimate of observed adult female survival.
\(\tau_t\)		Precision of Kaplan-Meier survival estimate.
\(\hat{I}_{i,t,m}\)	0 or 1	Indicator of whether animal \(i\) is dead or alive.
Estimated state
\(R_t\)	rate	Recruitment (calves per cow).
\(S_t\)	probability	Adult female survival.
\(N_t\)	animals	Adult female population size.
\(\bar{\lambda}_{y,t}\)	rate	Average population growth over \(y\) years.
\(X_t\)	rate	Recruitment adjusted for sex ratio and (optionally) delayed age of first reproduction.
\(W_t\)	animals	Surviving adult females before recruitment.
\(J_t\)	animals	New recruits to adult age class.
\(\lambda_t\)	rate	Annual population growth.
Estimated Parameters
\(\beta^R_0\)		Recruitment intercept.
\(\beta^R_a\)		Anthropogenic disturbance slope for recruitment.
\(\beta^R_f\)		Fire slope for recruitment.
\(\sigma^2_{R}\)		Variance in random effect of year on recruitment.
\(\beta^S_0\)		Survival intercept.
\(\beta^S_a\)		Anthropogenic disturbance slope for survival.
\(\sigma^2_{S}\)		Variance in random effect of year on survival.
True state
\(\dot{R}_t\)	rate	Recruitment (calves per cow).
\(\dot{S}_t\)	probability	Adult female survival.
\(\dot{N}_t\)	animals	Adult female population size.
\(\dot{X}_t\)	rate	Recruitment adjusted for sex ratio and (optionally) delayed age of first reproduction.
\(\dot{W}_t\)	animals	Surviving adult females before recruitment.
\(\dot{J}_t\)	animals	New recruits to adult age class.
True demographic parameters
\(N_0=5000\)	animals	Initial adult population size.
\(\bar{R}_t\)	rate	Expected recruitment (calves per cow).
\(\bar{S}_t\)	probability	Expected adult female survival.
\(\dot{\beta}^R_0\)		Recruitment intercept.
\(\dot{\beta}^R_a\)		Anthropogenic disturbance slope for recruitment.
\(\dot{\beta}^R_f\)		Fire slope for recruitment.
\(\dot{\beta}^S_0\)		Survival intercept.
\(\dot{\beta}^S_a\)		Anthropogenic disturbance slope for survival.
\(p_0=1\)		Maximum recruitment multiplier.
\(p_k=0.6\)		Recruitment multiplier at carrying capacity.
\(k=100\)		Carrying capacity multiplier.
\(b=4\)		Density dependence shape parameter.
\(a=1\)		Allee effect parameter.
\(r_{max}=1.3\)		Maximum population growth rate.
\(\nu_R=0.46\)		Coefficient of variation in \(\dot{R}_t\) among years.
\(\nu_S=0.08696\)		Coefficient of variation in \(\dot{S}_t\) among years.
\(l_R=0\)		Minimum value for \(\dot{R}_t\).
\(l_S=0\)		Minimum value for \(\dot{S}_t\).
\(h_R=0\)		Maximum value for \(\dot{R}_t\).
\(h_S=0\)		Maximum value for \(\dot{S}_t\).
Observation model parameters
\(d=1,2,4,8,16, \text{ or } 24\)	years	Monitoring duration.
\(n=15,30, \text{ or } 60\)	animals	Target number of collared females.
\(o=1 \text{ or } 4\)	years	Years between collar deployments.
\(y=3\)	years	Assessment period for population growth rate.
\(w=3,6,\text{ or } 9\)	ratio	The apparent number of adult females per collared animal in composition survey.
\(\dot{q}=\text{uniform}(0,0.6)\)	ratio	True ratio of young bulls to adult females in composition survey.
\(\dot{u}=\text{uniform}(0,0.2)\)	probability	True probability of misidentifying young bulls as adult females and vice versa in composition survey.
\(\dot{z}=\text{uniform}(0,0.2)\)	probability	True probability of missing calves in composition survey.
\(\dot{c}\)		True bias correction term in simulation model (Supplement B).
\(q=\text{uniform}(0,0.6)\)	ratio	Unknown ratio of young bulls to adult females in composition survey.
\(u=\text{uniform}(0,0.2)\)	probability	Unknown probability of misidentifying young bulls as adult females and vice versa in composition survey.
\(z=\text{uniform}(0,0.2)\)	probability	Unknown probability of missing calves in composition survey.
\(c\)		Unknown bias correction term in Bayesian model (Supplement B).
Expected value of perfect and sample information
\(\psi=\text{viable or not}\)	binary	True status of a population.
\(\alpha=\text{viable or not}\)	binary	Assigned population status.
\(\nu_{\alpha,\psi}=1 \text{ when } \alpha=\psi, 0 \text{ otherwise}\)	binary	Value of classification decision.
\(x\)		Simulated observations from an example population.
\(\tau\)		A disturbance and monitoring scenario.
\(P(\psi\|\tau)\)	probability	Prior probability of state \(\psi\).
\(P(\psi\|x,\tau)\)	probability	Posterior probability of state \(\psi\).
\(EVPI_\tau\)		Expected value of perfect information.
\(EVSI_\tau\)		Expected value of sample information.